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Lesson 10: What does f' say about f?

2.
Outline
Increasing and Decreasing functions
Concavity and the second derivative

3.
Deﬁnition
Let f be a function deﬁned on and interval I . f is called
increasing if
f (x1 ) < f (x2 ) whenever x1 < x2
for all x1 and x2 in I .

4.
Deﬁnition
Let f be a function deﬁned on and interval I . f is called
increasing if
f (x1 ) < f (x2 ) whenever x1 < x2
for all x1 and x2 in I .
f is called decreasing if
f (x1 ) > f (x2 ) whenever x1 < x2
for all x1 and x2 in I .

5.
Fact
If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b)

6.
Fact
If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b)
If f is decreasing on (a, b), then f (x) ≤ 0 for all x in (a, b).

7.
Fact
If f (x) > 0 for all x in (a, b), then f is increasing on (a, b).
If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).

8.
Outline
Increasing and Decreasing functions
Concavity and the second derivative

9.
Deﬁnition
A function is called concave up on an interval if f is
increasing on that interval.

10.
Deﬁnition
A function is called concave up on an interval if f is
increasing on that interval.
A function is called concave down on an interval if f is
decreasing on that interval.

11.
Fact
If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)

12.
Fact
If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
If f is concave down on (a, b), then f (x) ≤ 0 for all x in
(a, b).

13.
Fact
If f (x) > 0 for all x in (a, b), then f is concave up on (a, b).
If f (x) < 0 for all x in (a, b), then f is concave down on
(a, b).